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Full text of "Mathematical And Physical Papers - Iii"

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numerical calculation of a class of definite integrals and infinite series*.   The result is
I2            12.32    _    T.32.52
'     C\       A   I A .-- -A2         iS      T      />  / >4_____\3           * * "
I2   32           I2 32 5
These series, although ultimately divergent in all cases, are very convenient for numerical calculation when the modulus of mr is large. Moreover they give at once D' = 0 for the condition that FQ(r) shall not become infinite with r, and therefore we shall be able to obtain the required relation between 0 and D, provided we can express JD' as a function of G and D.
29. This may be effected by means of the integral of (85) expressed by different integrals. This form of the integral is already known. It becomes, by a slight transformation,
JF (r] = [  [C" + D" log (r sin2 co)} (emr cosw + e-wrcosw) ^w.. .(89), Jo
0", D" "being the two arbitrary constants. If we expand the exponentials in (89), and integrate the terms separately, we obtain, in fact, an expression of the same form as (87). This transformation requires the reduction of the definite integral
Pi =     cos2* ft) log sin to da.
If we integrate by parts, integrating cos co log sin co dco, and differentiating cos2*""1 co, we shall make P^ depend on P,._lt Assuming •P0 = QQ> pi = iQi- • •> aild generally
*"     2.4...2i we get
<?,= <2,-{2-' + 4-*...+ (2tn J=f log® - J
* See Ca??i6. P7iiL T^^is. Vol. ix. p. 182.   [4?tfe, Vol. n. p. 349.] t A demonstration "by Mr Ellis of the theorem
2 log sin6d6 = ~ log ft), o                        ^